8 research outputs found
Is symmetry identity?
Wigner found unreasonable the "effectiveness of mathematics in the natural
sciences". But if the mathematics we use to describe nature is simply a coded
expression of our experience then its effectiveness is quite reasonable. Its
effectiveness is built into its design. We consider group theory, the logic of
symmetry. We examine the premise that symmetry is identity; that group theory
encodes our experience of identification. To decide whether group theory
describes the world in such an elemental way we catalogue the detailed
correspondence between elements of the physical world and elements of the
formalism. Providing an unequivocal match between concept and mathematical
statement completes the case. It makes effectiveness appear reasonable. The
case that symmetry is identity is a strong one but it is not complete. The
further validation required suggests that unexpected entities might be
describable by the irreducible representations of group theory
Spontaneous symmetry breaking and bifurcations from the Maclaurin and Jacobi ellipsoids
The equilibrium of a rotating self-gravitating fluid is governed by non-linear equations. The equilibrium solutions, parametrized in terms of the angular momentum squared, exhibit the phenomenon of bifurcation, accompanied by spontaneous symmetry breaking. Under very general assumptions, a set of selection rules can be derived, which drastically restrict the patterns of symmetry breaking that are allowed to appear. Bifurcations of this kind are similar to second-order phase transitions à la Landau. The method is illustrated by the simple example of an incompressible fluid in rigid rotation. However, the selection rules are more general; they apply also to models which approximate a rotating star more realistically.L'état d'équilibre d'un fluide tournant soumis à son interaction gravitationnelle est déterminé par des équations non linéaires. Les solutions d'équilibre, paramétrées par le carré du moment cinétique, présentent des bifurcations accompagnées de brisures de symétrie. D'hypothèses très générales on déduit des règles de sélection concernant les brisures de symétrie qui peuvent apparaître dans ce problème. Les bifurcations sont du même type que celles à la Landau qui apparaissent dans les transitions de phase du second ordre. La méthode est illustrée par l'exemple simple d'un fluide incompressible animé d'une rotation globale et une nouvelle famille infinie de bifurcations est trouvée. Cependant les règles de sélection sont plus générales ; elles s'appliquent aussi aux modèles qui représentent la rotation d'une étoile de façon plus réaliste